Practical 8 - Distance Sampling

Aim of this practical:

In this practical we are going to look at distance sampling models.

Distance Sampling

In this practical we will:

• Fit a spatial distance sampling model
• Estimate animal abundance

Libraries to load:

library(dplyr)
library(INLA)
library(ggplot2)
library(patchwork)
library(inlabru)     
library(sf)
# load some libraries to generate nice map plots
library(scico)

The data

In the next exercise, we will explore data from a combination of several NOAA shipboard surveys conducted on pan-tropical spotted dolphins in the Gulf of Mexico. The data set is available in inlabru (originally obtained from the dsm R package) and contains the following information:

• A total of 47 observations of groups of dolphins were detected. The group size was recorded, as well as the Beaufort sea state at the time of the observation.

• Transect width is 16 km, i.e. maximal detection distance 8 km (transect half-width 8 km).

We can load and visualize the data as follows:

mexdolphin <- mexdolphin_sf

mexdolphin$depth <- mexdolphin$depth %>% mutate(depth=scale(depth)%>%c())

ggplot() + geom_sf(data = mexdolphin$points, color = "red" ) +

  geom_sf(data = mexdolphin$samplers) +

  geom_sf(data = mexdolphin$ppoly, alpha = 0)

The workflow

To model the density of spotted dolphins we take a thinned point process model of the form:

When fitting a distance sampling model we need to fulfill the following tasks:

1. Build the mesh

2. Define the SPDE representation of the spatial GF. This includes defining the priors for the range and sd of the spatial GF

3. Define the components of the linear predictor. This includes the spatial GF and all eventual covariates

4. Define the observation model using the bru_obs() function

5. Run the model using the bru() function

Building the mesh

The first task is to build the mesh that covers the area of interest. For this purpose we use the function fm_mesh_2d. To do so, we need to define the area of interest. We can either use a predefined boundary or create a non convex hull surrounding the location of the specie sightseeings

non-covex hull

boundary0 = fm_nonconvex_hull(mexdolphin$points,convex = -0.1)
mesh_0 = fm_mesh_2d(boundary = boundary0,
                          max.edge = c(30, 150), # The largest allowed triangle edge length.
                          cutoff = 15,
                          crs = fm_crs(mexdolphin$points))
ggplot() + gg(mesh_0)

domain boundary

The mexdolphin object contains a predefined region of interest which can be accessed through mexdolphin$ppoly

mesh_1 = fm_mesh_2d(boundary = mexdolphin$ppoly,
                    max.edge = c(30, 150),
                    cutoff = 15,
                    crs = fm_crs(mexdolphin$points))

ggplot() + gg(mesh_1)

Warning Task

Look at the documentation for the fm_mesh_2d function typing

?fm_mesh_2d

play around with the different options and create different meshes. You can compare these against a pre-computed mesh available by typing plot(mexdolphin$mesh)

The rule of thumb is that your mesh should be:

• fine enough to well represent the spatial variability of your process, but not too fine in order to avoid computation burden

• the triangles should be regular, avoid long and thin triangles.

• The mesh should contain a buffer around your area of interest (this is what is defined in the offset option) in order to avoid boundary artefact in the estimated variance.

Define the SPDE representation of the spatial GF

To define the SPDE representation of the spatial GF we use the function inla.spde2.pcmatern. This takes as input the mesh we have defined and the PC-priors definition for \(\rho\) and \(\sigma\) (the range and the marginal standard deviation of the field).

PC priors Gaussian Random field are defined in (Fuglstad et al. 2018). From a practical perspective for the range \(\rho\) you need to define two paramters \(\rho_0\) and \(p_{\rho}\) such that you believe it is reasonable that

\[ P(\rho<\rho_0)=p_{\rho} \]

while for the marginal variance \(\sigma\) you need to define two parameters \(\sigma_0\) and \(p_{\sigma}\) such that you believe it is reasonable that

\[ P(\sigma>\sigma_0)=p_{\sigma} \]

Tip Question

Take a look at the code below and select which of the following statements about the specified Matern PC priors are true.

spde_model <- inla.spde2.pcmatern(mexdolphin$mesh,
  prior.sigma = c(2, 0.01),
  prior.range = c(50, 0.01)
)

there is probability of 0.01 that the spatial range is greater or equal than 50 the probability that the spatial range is smaller than 50 is very small the probability that the marginal standard deviation is smaller than 2 is very small there is probability of 0.99 that the marginal standard deviation is less or equal than 2

Define the components of the linear predictor

We have now defined a mesh and a SPDE representation of the spatial GF. We now need to define the model components.

First, we need to define the detection function. Here, we will define a half-normal detection probability function. This must take distance as its first argument and the linear predictor of the sigma parameter as its second:

hn <- function(distance, sigma) {
  exp(-0.5 * (distance / sigma)^2)

}

We need to now separately define the components of the model including the SPDE model, the Intercept, the effect of depth and the detection function parameter sigma.

cmp <- ~ space(main = geometry, model = spde_model) +
  sigma(1,
    prec.linear = 1,
    marginal = bm_marginal(qexp, pexp, dexp, rate = 1 / 8)
  ) +
  Intercept(1)

Note

To control the prior distribution for the sigma parameter, we use a transformation mapper that converts a latent variable into an exponentially distributed variable with expectation 8 (this is a somewhat arbitrary value, but motivated by the maximum observation distance W)

The marginal argument in the sigma component specifies the transformation function taking N(0,1) to Exponential(1/8).

The formula, which describes how these components are combined to form the linear predictor

\[ \log \color{red}{\tilde{\lambda}(s)} = \overbrace{\log \lambda (s)}^{\beta_0 + \xi(s)} + \overbrace{\log \color{red}{g(d(s))}}^{-0.5~d(\mathbf{s})^2\sigma^{-2}} \]

Warning Task

Complete the code below to define the formula

eta <- ... + log(2)

eta <- geometry + distance ~ space +

  log(hn(distance, sigma)) +

  Intercept + log(2)

Here, the log(2) offset in the predictor takes care of the two-sided detections

Define the observation model

inlabru has support for latent Gaussian Cox processes through the cp likelihood family. To fit a point process model recall that we need to approximate the integral in using a numerical integration scheme as:

\[ \approx\exp\left(-\sum_{k=1}^{N_k}w_k\lambda(s_k)\right)\prod_{i=1}^n \lambda(\mathbf{s}_i) \]

Thus, we first create our integration scheme using the fm_int function by specifying integration domains for the spatial and distance dimensions.

Here we use the same points to define the SPDE approximation and to approximate the integral in ?@eq-thinned_pp, so that the integration weight and SPDE weights are consistent with each other. We also need to explicitly integrate over the distance dimension so we use the fm_mesh_1d() to create mesh over the samplers (which are the transect lines in this dataset, so we need to tell inlabru about the strip half-width).

# build integration scheme

distance_domain <-  fm_mesh_1d(seq(0, 8,
                              length.out = 30))

ips = fm_int(list(geometry = mexdolphin$mesh,
                  distance = distance_domain),
             samplers = mexdolphin$samplers)

Now, we just need to supply the sf object as our data and the integration scheme ips:

lik = bru_obs("cp",
              formula = eta,
              data = mexdolphin$points,
              ips = ips)

Then we fit the model, passing both the components and the observational model

fit = bru(cmp, lik)

Note

inlabru supports a shortcut for defining the integration points using the domain and samplers argument of bru_obs(). This domain argument expects a list of named domains with inputs that are then internally passed to fm_int() to build the integration scheme. The samplers argument is used to define subsets of the domain over which the integral should be computed. An equivalent way to define the same model as above is:

lik = bru_obs(formula = eta,
              data = mexdolphin$points,
              family = "cp",
              domain = list(
                geometry = mesh,
                distance = fm_mesh_1d(seq(0, 8, length.out = 30))),
              samplers = mexdolphin$samplers)

Visualize model Results

Posterior summaries

We can use the fit$summary.fixed and summary.hyperpar to obtain posterior summaries of the model parameters.

                  mean 0.025quant 0.975quant
sigma            -0.05      -0.46       0.36
Intercept        -8.16      -9.29      -7.34
Range for space 295.48     110.54     673.69
Stdev for space   0.81       0.42       1.39

Look at the SPDE parameter posteriors as follows:

plot( spde.posterior(fit, "space", what = "range")) +
plot( spde.posterior(fit, "space", what = "log.variance"))

Model predictions

We now want to extract the estimated posterior mean and sd of spatial GF.

Warning Task

Define a prediction grid using function fm_pixel(). Then compute the prediction for both the spatial GF and the linear predictor (spatial GF + intercept)

pxl <- fm_pixels(mexdolphin$mesh, dims = c(200, 100), mask = mexdolphin$ppoly)

pr.int = predict(fit, pxl, ~data.frame(spatial = space,

                                      lambda = exp(Intercept + space)))

Finally, we can plot the maps of the spatial effect

ggplot() + geom_sf(data = pr.int$spatial,aes(color = mean)) + scale_color_scico() + ggtitle("Posterior mean")

ggplot() + geom_sf(data = pr.int$spatial,aes(color = sd)) + scale_color_scico() + ggtitle("Posterior sd")

Note The posterior sd is lowest at the observation points. Note how the posterior sd is inflated around the border, this is the “border effect” due to the SPDE representation.

Warning Task

Using the predictions stored in pr.int, produce a map of the posterior mean intensity.

Recall that the predicted intensity is given by \(\lambda(s) = \exp(\beta_0+\xi(s))\)

ggplot() +

  geom_sf(data = pr.int$lambda,aes(color = mean)) +

  scale_color_scico(palette = "imola") +

  ggtitle("Posterior mean")

We can predict the detection function in a similar fashion.Here, we should make sure that it doesn’t try to evaluate the effects of components that can’t be evaluated using the given input data.

distdf <- data.frame(distance = seq(0, 8, length.out = 100))

dfun <- predict(fit, distdf, ~ hn(distance, sigma))

plot(dfun)

Abundance estimates

The mean expected number of animals can be computed by integrating the intensity over the region of interest as follows:

predpts <- fm_int(mexdolphin$mesh, mexdolphin$ppoly)
Lambda <- predict(fit, predpts, ~ sum(weight * exp(space + Intercept)))
Lambda

      mean       sd   q0.025     q0.5   q0.975   median sd.mc_std_err
1 250.0936 55.70774 147.5369 240.1456 373.4508 240.1456      4.763493
  mean.mc_std_err
1        6.523473

To fully propagate the uncertainty on the expected number animals we can draw Monte Carlo samples from the fitted model as follows (this could take a couple of minutes):

Ns <- seq(50, 450, by = 1)

Nest <- predict(fit, predpts,
  ~ data.frame(
    N = Ns,
    density = dpois(
      Ns,
      lambda = sum(weight * exp(space + Intercept))
    )
  ),
  n.samples = 2000
)

We can compare this with a simpler “plug-in” approximation:

Nest <- dplyr::bind_rows(
  cbind(Nest, Method = "Posterior"),
  data.frame(
    N = Nest$N,
    mean = dpois(Nest$N, lambda = Lambda$mean),
    mean.mc_std_err = 0,
    Method = "Plugin"
  )
)

Then, we can visualize the result as follows:

ggplot(data = Nest) +
  geom_line(aes(x = N, y = mean, colour = Method)) +
  geom_ribbon(
    aes(
      x = N,
      ymin = mean - 2 * mean.mc_std_err,
      ymax = mean + 2 * mean.mc_std_err,
      fill = Method,
    ),
    alpha = 0.2
  ) +
  geom_line(aes(x = N, y = mean, colour = Method)) +
  ylab("Probability mass function")